The minimal logarithmic signature conjecture states that in any finite simple group there are subsets A i , 1 ≤ i ≤ k such that the size |A i | of each A i is a prime or 4 and each element of the group has a unique expression as a product k i=1 x i of elements x i ∈ A i . The conjecture is known to be true for several families of simple groups. In this paper the conjecture is shown to be true for the groups − 2m (q), + 2m (q), when q is even, by studying the action on suitable spreads in the corresponding projective spaces. It is also shown that the method can be used for the finite symplectic groups. The construction in fact gives cyclic minimal logarithmic signatures in which each A i is of the form {y j i | 0 ≤ j < |A i |} for some element y i of order ≥ |A i |.

A logarithmic signature (LS) for a finite group G is an ordered tuple α = [A 1 , A 2 , . . . , A n ] of subsets A i of G, such that every element g ∈ G can be expressed uniquely as a product g = a 1 a 2 · · · a n , where a i ∈ A i . The length of an LS α is defined to be l(α) = n i=1 |A i |. It can be easily seen that for a group G of order k j=1 p j m j , the length of any LS α for G, satisfies, l(α) ≥ k j=1 m j p j . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (González Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002. The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families P SL n (q) and P Sp 2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.